The present invention relates to devices and methods for compensating for the Gouy phase shift in quasi-phase matched optical nonlinear processes.
Nonlinear optical devices are very useful for converting laser light between different wavelengths. Examples include devices for second harmonic generation (SHG) in which two photons are combined to create a new photon at twice the frequency (half the wavelength), and devices for optical parametric generation in which a single photon splits into two photons at longer wavelengths. In general, to obtain high conversion efficiencies for these interactions it is necessary to phase match the various propagating waves to allow for continuous addition of the generated light intensity. The requirement for phase matching is that the phase velocities of the interacting waves in the nonlinear material must be equal. However, the naturally occurring dispersion present in all materials means that the phase velocities (or equivalently refractive indices) will not be equal at the various wavelengths of interest. Therefore, various non-trivial schemes are used to either achieve phase matching, or to overcome the problems of imperfect phase matching. A conventional phase-matching technique makes use of birefringence in a crystalline material; birefringence and different interacting optical polarisation states can be used in combination to achieve efficient interactions.
An alternative phase matching technique is that of quasi-phase matching (QPM), in which the difference in phase velocities of the interacting waves is compensated by a periodic reversal of the nonlinear coefficient of the crystal along the propagation direction. In this scheme, the phase velocities are not equalised, but the periodic reversal of the nonlinear coefficient overcomes the deleterious effects of the absence of phase matching. The periodic reversal is commonly referred to as a grating.
A further requirement to obtain high conversion efficiency is the use of high power fundamental input beams, since nonlinear conversion efficiency is generally proportional to the square of the fundamental power. High power beams can be achieved by confinement of the beam dimensions through focusing in bulk materials, or alternatively with the use of waveguide devices. Waveguides provide an efficient route towards high conversion efficiencies even at lower fundamental input powers but, due to their limited power handling capabilities, are less suited for higher power regimes, for which bulk-focused interactions are essential.
However, for focused laser beam interactions there is an effect known as the Gouy shift [1]. This occurs with all focused optical beam interactions, and is a phase shift occurring whenever a beam passes through a focus. The phase shift causes a slight increase in the spatial frequency of phase fronts for the focused wave compared to a simple plane wave. For a single mode Gaussian beam, as is common for laser output, this position dependent phase shift is seen as a π advancement of the phase fronts of the propagating beam as it travels from −∞ to +∞ through a focused waist. The effect is an inevitable consequence of focusing, and causes a slight phase mismatch whenever a focused beam is used in a nonlinear material. Thus, perfect phase matching along the length of a nonlinear device is prevented.
Previous work by Boyd and Kleinman [2] has found the conditions which provide optimal conversion efficiency under the effects of focusing in bulk nonlinear devices, by optimisation of the focusing parameter, specifically the focused Rayleigh range, and the nonlinear material length. Their analysis, for a birefringent interaction, assumes a linearly invariant device and optimises the focusing parameters accordingly, with the aim of achieving a balance between tight focusing to give a high power at the beam waist and the need to utilise as much interaction length as possible. However, with the limitation of a linearly invariant device, Boyd and Kleinman were not able to compensate for the Gouy shift. Instead they chose to optimise to a focusing value that minimises its deleterious effects on phase matching whilst attempting to maximise the benefits of tight focusing.
The optimisation provided by Boyd and Kleinman has also proven valid for focused interactions in periodic QPM grating structures. These devices are similarly linearly invariant, so that as with a birefringently phase matched device, perfect phase matching under focusing is prevented.
Hence, for both birefringent phase matching and quasi-phase matching, the Gouy phase shift is detrimental to the conversion efficiency of the nonlinear interaction. The Gouy shift is often ignored, however.